Are graph transformations on the Foundation maths paper?

No. Spec ref A13, sketching translations and reflections of the graph of a given function, is Higher tier only in Edexcel 1MA1. Foundation candidates meet the shapes of quadratic, cubic and reciprocal graphs, but they are never asked what happens to a curve when it becomes f(x + 2) or -f(x).

Why does f(x + 2) move the graph left instead of right?

Because the 2 is added to the input before the function acts on it. The new curve reaches a given height as soon as x + 2 equals the old x-value, which happens 2 units earlier, so the whole graph arrives 2 units sooner and sits 2 to the left. Reading it the other way round, f(x - 2) moves the graph 2 units to the right.

Do I need stretches such as 2f(x) and f(2x) for Edexcel GCSE?

Not for 1MA1. The A13 wording covers translations and reflections of a given function, and stretches are A level content. What Edexcel does set at Higher is a pair of these four applied together, for example the maximum point of y = -f(x) + 5, so practise tracking a point through two steps in order.

How many marks are graph transformations worth on Edexcel Higher?

Usually 2-4 marks on a paper, and they are cheap marks. The write-down parts, giving the image of a maximum point or naming a transformation, are worth 1 mark each and take seconds once the direction rules are secure. The longer version asks for the equation of a transformed curve or for a point tracked through two transformations.

Graph Transformations - Edexcel GCSE Higher Maths

Typically 2-4 marks per paper, usually as one-mark write-down parts · Spec 1MA1

Spec point A13 asks you to sketch translations and reflections of the graph of a given function, and in 1MA1 it belongs to Higher tier only. Four changes to $y = f(x)$ do all of the work. Writing $f(x) + a$ slides the curve $a$ units up, $f(x + a)$ slides it $a$ units to the left, $-f(x)$ reflects it in the $x$ -axis, and $f(-x)$ reflects it in the $y$ -axis. The dividing line is the function bracket: a change written outside it acts on the $y$ -coordinates and does what it appears to say, while a change written inside acts on the $x$ -coordinates and does the opposite.

The useful thing about A13 is that you are almost never told what $f(x)$ actually is. A standard Edexcel stem gives the curve $y = f(x)$ with its maximum point at $(3, 5)$ and then asks, one mark at a time, for the maximum point of $y = f(x) + 2$ , of $y = f(x - 3)$ and of $y = -f(x)$ . The other regulars are describe fully the single transformation that maps $y = f(x)$ onto $y = f(x) + 7$ , write down an equation for a curve after a stated translation, and the trigonometric version, where you are handed $y = \sin x^\circ$ and asked where the maximum of $y = \sin(x + 30)^\circ$ sits between $0$ and $360$ . It can land on any of the three papers, and a calculator changes nothing, because every step is bookkeeping on a pair of coordinates.

Marks go in two places. The first is the direction of a horizontal shift: $f(x + 3)$ moves the curve left, and a candidate who moves it right has lost the mark on the opening line with nothing to recover. The safest route is to set the bracket equal to the old $x$ -value and solve, so for $f(x + 3)$ the old point at $x = 2$ reappears where $x + 3 = 2$ , at $x = -1$ . The second is the wording of a description. Edexcel pays for the name of the transformation together with the detail: translation plus a column vector or a distance and a direction, reflection plus the equation of the mirror line. Stretches such as $2f(x)$ and $f(2x)$ sit outside the A13 wording and belong to A level, so the four transformations above are the whole list you need.

Worked example

The curve with equation

y = f(x)

has exactly one turning point, a minimum at

(4, -3)

. (a) Write down the coordinates of the minimum point of the curve with equation

y = f(x) + 7

. (1 mark) (b) Write down the coordinates of the minimum point of the curve with equation

y = f(x - 2)

. (1 mark) (c) Write down the coordinates of the turning point of the curve with equation

y = -f(x)

, and state whether it is a maximum or a minimum. (2 marks) (d) The curve with equation

y = f(x)

is transformed to the curve with equation

y = f(x) - 9

. Describe fully the single transformation. (1 mark)

[5 marks]

(a) The $+7$ sits outside the bracket, so it acts on the outputs: every point rises 7 and nothing moves sideways. The minimum is at $(4, 4)$ .
B1: $(4, 4)$ ; a vertical translation leaves the $x$ -coordinate alone
(b) The $-2$ sits inside the bracket, so the movement is horizontal and runs opposite to the sign. Set $x - 2 = 4$ , giving $x = 6$ , so the minimum is at $(6, -3)$ .
B1: $(6, -3)$ ; $(2, -3)$ is the standard wrong answer, from moving left instead of right
(c) $-f(x)$ multiplies every output by $-1$ , which is a reflection in the $x$ -axis, so $(4, -3)$ lands at $(4, 3)$ .
B1: $(4, 3)$ ; only the $y$ -coordinate changes sign
(c) The whole curve is flipped over the $x$ -axis, so the lowest point of $y = f(x)$ becomes the highest point of $y = -f(x)$ . It is a maximum.
B1: maximum, awarded on a correct reflection
(d) Subtracting 9 outside the bracket drops every point by 9: a translation of 9 units in the negative $y$ -direction, that is by the vector $\binom{0}{-9}$ .
B1: the name and the detail together; the word translation on its own scores nothing

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no grids here, so each curve is pinned down by its equation or by named points on it, which is also how the write-down parts of a real paper work once the sketch is out of the way. Coordinate answers take two boxes, $x$ first, and negative values are typed with a minus sign. The trigonometric questions all use degrees in the interval $0$ to $360$ , and none of them needs a calculator.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [1 mark]

The graph of $y = f(x)$ passes through the point $(3, 8)$ . Write down the coordinates of the corresponding point on the graph of $y = f(x) + 4$ .

x = y =
Question 2 Warm-up [1 mark]

The graph of $y = f(x)$ passes through the point $(2, 7)$ . Write down the coordinates of the corresponding point on the graph of $y = -f(x)$ .

x = y =
Question 3 Warm-up [1 mark]

The graph of $y = f(x)$ passes through the point $(4, 9)$ . Write down the coordinates of the corresponding point on the graph of $y = f(x - 3)$ .

x = y =
Question 4 Exam pace [2 marks]

The graph of $y = f(x)$ passes through the point $(7, 3)$ . The graph of $y = f(x + a)$ passes through the point $(2, 3)$ . Find the value of $a$ .

Your answer (fractions like 1/2 are fine)
Question 5 Stretch [2 marks]

The curve with equation $y = f(x)$ has exactly one turning point, a maximum at $(3, 5)$ . Find the coordinates of the maximum point of the curve with equation $y = f(x - 2) + 6$ .

x = y =
Question 6 Stretch [2 marks]

The curve with equation $y = f(x)$ has a maximum point at $(-3, 8)$ . The curve with equation $y = f(x - c)$ has its maximum point at $(4, 8)$ . Find the value of $c$ .

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Reading $f(x + 3)$ as a shift to the right, so the point $(2, 5)$ is handed in as $(5, 5)$ when its image is $(-1, 5)$ .

Set the bracket equal to the old $x$ -value and solve it. The new curve repeats the old height where $x + 3 = 2$ , so $x = -1$ : inside the bracket, the graph travels against the sign.
Swapping $-f(x)$ and $f(-x)$ , so a maximum at $(4, 6)$ is reported as $(-4, 6)$ under $y = -f(x)$ .

Decide which coordinate the minus sign reaches. Outside the bracket it is applied after the function, so it negates the output and $(4, 6)$ becomes $(4, -6)$ . Inside the bracket it negates the input, and only then does the $x$ -coordinate flip.
Moving both coordinates for a single transformation, giving the image of $(2, 7)$ under $y = f(x) + 4$ as $(6, 11)$ .

One of these transformations touches one coordinate. Write the pair down, cross out the coordinate that cannot move, then do the arithmetic on the other one.
Answering describe fully with the single word translation, or with it goes down 3, or with a pair of transformations when the question asked for one.

Give the name and the numbers in the same sentence: translation by the column vector 0 across and 3 down, or translation of 3 units in the negative $y$ -direction. For a reflection, name the mirror line, so reflection in the $x$ -axis rather than reflected.
Transforming an equation by tacking the number on the end: with $f(x) = x^2 + 4x$ , writing $f(x - 3) = x^2 + 4x - 3$ .

Replace every $x$ in the expression by $x - 3$ , brackets and all, so $f(x - 3) = (x - 3)^2 + 4(x - 3)$ . What you wrote instead was $f(x) - 3$ , a curve translated down rather than across.

Frequently asked questions

Are graph transformations on the Foundation maths paper?: No. Spec ref A13, sketching translations and reflections of the graph of a given function, is Higher tier only in Edexcel 1MA1. Foundation candidates meet the shapes of quadratic, cubic and reciprocal graphs, but they are never asked what happens to a curve when it becomes $f(x + 2)$ or $-f(x)$ .
Why does f(x + 2) move the graph left instead of right?: Because the 2 is added to the input before the function acts on it. The new curve reaches a given height as soon as $x + 2$ equals the old $x$ -value, which happens 2 units earlier, so the whole graph arrives 2 units sooner and sits 2 to the left. Reading it the other way round, $f(x - 2)$ moves the graph 2 units to the right.
Do I need stretches such as 2f(x) and f(2x) for Edexcel GCSE?: Not for 1MA1. The A13 wording covers translations and reflections of a given function, and stretches are A level content. What Edexcel does set at Higher is a pair of these four applied together, for example the maximum point of $y = -f(x) + 5$ , so practise tracking a point through two steps in order.
How many marks are graph transformations worth on Edexcel Higher?: Usually 2-4 marks on a paper, and they are cheap marks. The write-down parts, giving the image of a maximum point or naming a transformation, are worth 1 mark each and take seconds once the direction rules are secure. The longer version asks for the equation of a transformed curve or for a point tracked through two transformations.

Graph Transformations - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics